P7.1SOLUTION
A.Using the assumption of annual compounding,
Et=E0(1 + g)t
420=311(1 + g)10
1.35=(1 + g)10
ln(1.35)=10 ln(1 + g)
0.300/10=ln(1 + g)
e0.030=1 + g
1.031 - 1=g
g=0.031 or 3.1%
Using the continuous compounding assumption,
Et=E0egt
420=311e10g
1.35=e10g
ln(1.35)=10g
g=0.3000/10
=0.03 or 3.00%
Using the same methods, continuous growth model estimates for various occupations are:
Employment (1,000) / Continuous Growth ModelOccupation / 1998 / 2008 / Annual Compounding / Continuous Compounding
Bill collectors / 311 / 420 / 3.05% / 3.00%
Computer engineers / 299 / 622 / 7.60% / 7.32%
Physicians assistants / 66 / 98 / 4.03% / 3.95%
Respiratory therapists / 86 / 123 / 3.64% / 3.57%
Systems analysts / 617 / 1,194 / 6.82% / 6.60%
B.For example, if the number of jobs jumps to 420,000 from 311,000 over a ten-year period, then a 3.05% rate of job growth is indicated when annual compounding is assumed. With continuous compounding, a 3.00% rate of growth leads to a similar growth in jobs over a ten-year period. Of course, this small difference is due to the amount of Ainterest-on-interest.@ Either method can be employed to measure the rate of growth, but managers must make growth comparisons using a consistent basis.
P7.6SOLUTION
A.
St+1=St +
-
=St +
-
=
- 2.5St
B.
St+1=2($500,000)(1.02) + 2($500,000)(0.80)
- 0.5 ($500,000)(1.10) - 2.5 ($500,000)
=$1,020,000 + $800,000 - $275,000 - $1,250,000
=$295,000
P7.8SOLUTION
A.St+1=St + ΔS
=St - ΔSP + ΔST
=St - 2(Pt+1/Pt - 1)St + 3(Tt+1/Tt - 1)St
=-2(Pt+1/Pt)St + 3(Tt+1/Tt)St
B.St+1=-2(16.5/15)10,000 + 3(1.15)10,000
=-22,000 + 34,500
=12,500 games
8.4 C.Initially, let A = B = 100, so output is:
Q = 4(100) + 6(100) + 8(100)(100) = 81,000
Increasing both inputs by an arbitrary percentage, say, 1%, leads to:
Q = 4(101) + 6(101) + 8(101)(101) = 82,618
Because a 1% increase in both inputs results in a 2% increase in output (Q2/Q1 = 82,618/81,000 = 1.02), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale.
D.Initially, let L = K = 100, so output is:
Q = 7(1002) + 5(100)(100) + 2(1002) = 140,000
Increasing both inputs by an arbitrary percentage, say, 2%, leads to:
Q = 7(1022) + 5(102)(102) + 2(1022) = 145,656
Because a 2% increase in both inputs results in a 4% increase in output (Q2/Q1 = 145,656/140,000 = 1.04), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale.
P8.7SOLUTION
A.Because Route 66 operates in a perfectly competitive industry, the 104 price premium for full-service versus self-service gasoline is stable. Thus, the net marginal revenue product of attendant labor (sometimes referred to as the value of marginal product) is:
Route 66 Truck Stop, Inc.Number of
Attendants
per Day
(1) / Full Service
Output
(gallons)
(2) / Marginal
Product
of Labor
(3) / Net Marginal
Revenue
Product of Labor
(4) = (3) 104
0 / 0 / -- / --
1 / 2,000 / 2,000 / $200
2 / 3,800 / 1,800 / 180
3 / 5,400 / 1,600 / 160
4 / 6,800 / 1,400 / 140
5 / 8,000 / 1,200 / 120
B.From the table above, it becomes clear that employment of three attendants could be justified at a daily wage cost of $160 because MRPA=3 = $160. Employment of a fourth attendant could not be justified because MRPA=4 = $140 < $160.
C.From the table above, the marginal revenue product of a fourth attendant MRPA=4 = $140. Thus, $140 is the highest daily wage cost Route 66 would be willing to pay to hire a staff of four attendants.
P9.1SOLUTION
A.True. The point of minimum average cost identifies the minimum efficient scale of plant. By definition, average and marginal costs are equal at this point.
B.False. The breakeven activity level is where Q = TFC/(P - AVC). As average variable cost (AVC) increases, this ratio and the breakeven activity level will also increase.
C.True. When εC > 1, the percentage change in cost exceeds a given percentage change in output. This describes a situation of increasing average costs and diseconomies of scale.
D.True. When long-run average costs are declining, it can pay to operate larger plants with some excess capacity rather than smaller plants at their peak efficiency.
E.False. The degree of operating leverage is defined DOL = Q(P - AVC)/(Q(P - AVC) - TFC). Therefore, when total fixed costs are zero, DOL is a constant and an increase in average variable cost will have no effect on DOL.